Barbara M. Maenhaut

Decompositions into 2-regular subgraphs and equitable partial cycle decompositions

Two theorems are proved in this paper. Firstly, it is proved that there exists a decomposition of the complete graph of order n into t edge-disjoint 2-regular subgraphs of orders m1, m2,...,mt if and only if n is odd, 3≤mi ≤ n for i = 1, 2,...,t, and m1 + m2 +...+ mt = (n 2). Secondly, it is proved that if there exists partial decomposition of the complete graph Kn of order n into t cycles of lengths m1, m2,..., mt, then there exists an equitable partial decomposition of Kn into t cycles of lengths m1, m2,..., mt. A decomposition into cycles is equitable if for any two vertices u and v, the number of cycles containing u and the number of cycles containing v differ by at most 1.

A Family of Perfect Factorisations of Complete Bipartite Graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p2 for an odd prime p. We construct a family of (p?1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cycle.

New families of atomic Latin squares and perfect 1-factorisations

A perfect 1-factorisation of a graph G is a decomposition of G into edge disjoint 1-factors such that the union of any two of the factors is a Hamiltonian cycle. Let p ≥ 11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of Kp+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of Kp, p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of Kp, p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.

Existence and embeddings of partial Steiner triple systems of order ten with cubic leaves

Denote the set of 21 non-isomorphic cubic graphs of order 10 by L. We first determine precisely which L@?L occur as the leave of a partial Steiner triple system, thus settling the existence problem for partial Steiner triple systems of order 10 with cubic leaves. Then we settle the embedding problem for partial Steiner triple systems with leaves L@?L. This second result is obtained as a corollary of a more general result which gives, for each integer v>=10 and each L@?L, necessary and sufficient conditions for the existence of a partial Steiner triple system of order v with leave consisting of the complement of L and v-10 isolated vertices.

Decompositions of complete multigraphs into cycles of varying lengths

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

Nonextendible Latin Cuboids

We show that for all integers

Subsquare-free Latin squares of odd order

m \geqslant 4

On the Volume of 5-Cycle Trades

there exists a

On the Volume of 4-Cycle Trades

2m\times 2m\times m

On the non-existence of pair covering designs with at least as many points as blocks

latin cuboid that cannot be completed to a